Does "be engaged in" mean "have the symptoms of " here?

First the University of Minnesota neonatal-pediatrics professor Dana Johnson shared photos and videos that he’d collected in Romania of rooms teeming with children engaged in “motor stereotypies”: rocking, banging their heads, squawking.

Is it common to use "be engaged in" to describe people’s symptoms?

Solution 1:

No. "Motor stereotypies" are the symptoms, and the children are doing ("engaged in") them.

Consider if you would say the children "have the symptoms" of rocking, banging, squawking - you absolutely would not. "motor stereotypies" is simply a collective term for these behaviours.

"engaged in" cannot be used to describe symptoms such as a rash or sore throat. It is not uncommon as a way of saying someone is carrying out activities though.

Solution 2:

Answering the question in the title: No, because "motor stereotypies" is not a disease, which would need to be the case if the sentence read

... rooms teeming with children having the symptoms of “motor stereotypies": rocking, banging their heads, squawking.

The "motor stereotypies", which are then listed, are themselves symptoms so the sentence now effectively says "symptoms of symptoms".

Lexico say

engaged
ADJECTIVE

1 Busy; occupied.
The building is packed with people busily engaged in conversation.

So the original sentence means that the children are preoccupied with the behaviour described. Answering the question at the bottom, this is a normal way to phrase something. You would need to say what those actions are symptoms of, when describing them as "symptoms". You can't say "these are symptoms" without reference to the disease of which they are symptomatic.